Juergen Bokowski also sent me an article "Die Geschichte eines Modells"
in which he describes the difficult path involving the construction of
many temporary models to find a good shape for the surface and pleasing
placement of the vertices and edges, and some
other
publications.The solution above corresponds to a model with D6 symmetry (one C3
and three C2 axes). It is this topological configuration for which he could
first prove that a corresponding polyhedral object with 12 vertices, 66
straight edges, and 44 plane triangles could not be constructed without
self-intersections. Juergen Bokowski got me involved in this project by
asking me whether I could see a way to make a good computer graphics model
of such a mathematical object which would nicely show the symmetry and
the 44 curved triangular surface patches.

The genus-6 surface can be made of even higher symmetry, i.e. T24, corresponding
to the symmetry group of a tetrahedron, and the K-12 graph can be mapped
onto it with T12 symmetry, corresponding to an oriented tetrahedron.
I was unable to figure out how to place the 66 edges onto such a surface
by simply making diagrams or perspective sketches in 2 dimensions; I also
needed a physical 3D model to experiment. After contemplating many different
materials, I finally used tubular styrofoam pieces, 2 inches in diameter,
that are sold by hardware stores as insulating sleeves for warm-water pipes.
Twenty-two segments form a 3D graph with five four-fold junction points
at the locations of the corners and center of a tetrahedron, and with six
handles corresponding to its edges. At a diameter of 14 inches, the model
was easy to handle, yet open enough so that I could comfortably access
all points on its surface.

First I contemplated using marker pens or wires to show the placement
of the edges, but I then used thin strips of colored adhesive tape. On
the styrofoam surface, they can be applied and removed easily, and their
bright colors make it easy to track edges that wind around one of the tubular
segments. Randomly placing 12 vertices and then trying to connect them
could end up to be a very tedious task. Judicious use of symmetry not only
makes this task much easier, but it also leads to a more pleasing and easier
to understand model. The 12 vertices could either be placed as 6 pairs
in a C2-symmetric configuration around the centers of the 6 handles, or
they could be placed in a C3-symmetric manner around each of the 4 arms
of the central tetrahedral junction. I chose to place them at the 12 monkey
saddle points where two handle surfaces meet with one arm from the central
junction:

This gives the most room around each vertex to place the eleven emerging
edges.

Further contemplation of the symmetry of this object made it clear that
each of the eight C3-axis points on the surface had to correspond to the
center of three-fold symmetrical triangular face -- possibly drawn out
into an extended and warped "starfish" shape. Moreover, three different
types of C2-symmertical edges had to run through each of the 18 surface
points where the three C3 axis penetrated. While this placed many constraints
on the edges, it still took me three tries to find a scheme that would
properly connect every vertex with evey other one, and then two more refinements
to make the edges look as simple and pleasing as possible. The key degree
of freedom here is the amount of twist that one assigns to the edges on
the four branches of the central tetrahedral junction. First I started
here with the simplest possible pattern, which then forced more twist onto
the already rather long edges spiralling along the outer six handles. I
then realized, that the trick is to make the outer four C3-triangles (outlined
by the green edges) as simple as possible, and then pick up whatever twist
is necessary on the inner four branches. The most twisted inner edges (shown
in black), then each make two 180-degree turns around their respective
branches. All but one set of 12 triangles, can now be traced rather effortlessly
by visual inspection from a static viewpoint.

Now the challenge is how to convert this into beautiful computer graphics
model ...